10.1007/s40096-022-00504-x

The application of fuzzy transform method to the initial value problems of linear differential–algebraic equations

  • S. Mirzajani*1,
  • F. Bahrami1,
  • S. Shahmorad1,
  1. Department of Applied Mathematics, Faculty of Mathematical Science, University of Tabriz, Tabriz, IR

Published in Issue 2022-12-30

How to Cite

Mirzajani, S., Bahrami, F., & Shahmorad, S. (2022). The application of fuzzy transform method to the initial value problems of linear differential–algebraic equations. Mathematical Sciences, 18(2 (June 2024). https://doi.org/10.1007/s40096-022-00504-x
Crossref
Scopus
Google Scholar
Europe PMC

Abstract

Abstract In this study, we propose a method based on the direct and inverse fuzzy transforms ( F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {F}}$$\end{document} -transforms) to approximate solutions of the system of linear differential–algebraic equations. We obtain an analytical solution to the initial value problem in terms of basic functions using this method. Since the basic functions have compact support, by employing fuzzy transform to the problem, we obtain the system of linear algebraic equations with lower Hessenberg coefficient matrix. We discuss the solvability of this system and give the error analysis using the consistency of the initial value. We present several examples to illustrate the efficiency and the performance of this new method.

Keywords

  • Linear differential–algebraic equations,
  • Fuzzy partition,
  • F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {F}}$$\end{document}-transforms

References

  1. Alikhani et al. (2017) Trigonometric Fmdocumentclass[12pt]{minimal}
  2. usepackage{amsmath}
  3. usepackage{wasysym}
  4. usepackage{amsfonts}
  5. usepackage{amssymb}
  6. usepackage{amsbsy}
  7. usepackage{mathrsfs}
  8. usepackage{upgreek}
  9. setlength{oddsidemargin}{-69pt}
  10. begin{document}$$F^{m}$$end{document}-transform and its approximation properties 21(13) (pp. 3567-3577) https://doi.org/10.1007/s00500-017-2637-0
  11. Ascher et al. (1995) Stabilization of constrained mechanical systems with DAEs and invariant manifolds 23(2) (pp. 135-157)
  12. Ascher and Petzold (1998) SIAM https://doi.org/10.1137/1.9781611971392
  13. Di Martino et al. (2008) An image coding/decoding method based on direct and inverse fuzzy transforms 48(1) (pp. 110-131) https://doi.org/10.1016/j.ijar.2007.06.008
  14. Di Martino et al. (2010) Fuzzy transforms method and attribute dependency in data analysis (pp. 493-505) https://doi.org/10.1016/j.ins.2009.10.012
  15. Gaeta et al. (2015) Multisignal 1-D compression by Fdocumentclass[12pt]{minimal}
  16. usepackage{amsmath}
  17. usepackage{wasysym}
  18. usepackage{amsfonts}
  19. usepackage{amssymb}
  20. usepackage{amsbsy}
  21. usepackage{mathrsfs}
  22. usepackage{upgreek}
  23. setlength{oddsidemargin}{-69pt}
  24. begin{document}$$F$$end{document}-transform for wireless sensor networks applications (pp. 329-340) https://doi.org/10.1016/j.asoc.2014.11.061
  25. Gaeta et al. (2016) Cubic B-spline fuzzy transforms for an efficient and secure compression in wireless sensor networks (pp. 19-30) https://doi.org/10.1016/j.ins.2015.12.026
  26. Ghaneai and Hosseini (2016) Solving differential-algebraic equations through variational iteration method with an auxiliary parameter 40(5–6) (pp. 3991-4001) https://doi.org/10.1016/j.apm.2015.10.002
  27. Hosseini (2006) Adomian decomposition method for solution of nonlinear differential algebraic equations 181(2) (pp. 1737-1744)
  28. Jafari and Aminataei (2011) Solution of differential algebraic equations via semi-analytic method 2(2) (pp. 119-124) https://doi.org/10.1016/j.asej.2011.07.004
  29. Khastan et al. (2016) A new fuzzy approximation method to Cauchy problems by fuzzy transform (pp. 75-95) https://doi.org/10.1016/j.fss.2015.01.001
  30. Lamour et al. (2013) Springer https://doi.org/10.1007/978-3-642-27555-5
  31. Mehrmann and Kunkel (2006) European Mathematical Society
  32. Novák et al. (2008) Mining pure linguistic associations from numerical data 48(1) (pp. 4-22) https://doi.org/10.1016/j.ijar.2007.06.005
  33. Novák et al. (2013) F-transform and Fuzzy natural logic in time series analysis (pp. 40-47)
  34. Perfilieva, I.: Fuzzy transform: application to the reef growth problem. In: Fuzzy Logic in Geology, pp. 275–300 (2004)
  35. Perfilieva (2006) Fuzzy transforms: theory and application 157(8) (pp. 993-1023) https://doi.org/10.1016/j.fss.2005.11.012
  36. Perfilieva (2007) Fuzzy transforms: a challenge to conventional transforms (pp. 137-196) https://doi.org/10.1016/S1076-5670(07)47002-1
  37. Perfilieva, I., Daňková, M.: Image fusion on the basis of fuzzy transforms. In: Computational Intelligence in Decision and Control, pp. 471–476 (2008)
  38. Perfilieva et al. (2011) Toward a higher degree Fdocumentclass[12pt]{minimal}
  39. usepackage{amsmath}
  40. usepackage{wasysym}
  41. usepackage{amsfonts}
  42. usepackage{amssymb}
  43. usepackage{amsbsy}
  44. usepackage{mathrsfs}
  45. usepackage{upgreek}
  46. setlength{oddsidemargin}{-69pt}
  47. begin{document}$$F$$end{document}-transform (pp. 3-19) https://doi.org/10.1016/j.fss.2010.11.002
  48. Perfilieva, I., De Meyer, H., De Baets, B., Piskova, D.: Cauchy problem with fuzzy initial condition and its approximate solution with the help of fuzzy transform. In: IEEE International Conference on Fuzzy Systems (IEEE World Congress on Computational Intelligence), pp. 2285–2290 (2008)
  49. Soltanian et al. (2009) Application of He’s variational iteration method for solution of differential-algebraic equations 41(1) (pp. 436-445) https://doi.org/10.1016/j.chaos.2008.02.004
  50. Zeinali et al. (2018) On the structural properties of Fmdocumentclass[12pt]{minimal}
  51. usepackage{amsmath}
  52. usepackage{wasysym}
  53. usepackage{amsfonts}
  54. usepackage{amssymb}
  55. usepackage{amsbsy}
  56. usepackage{mathrsfs}
  57. usepackage{upgreek}
  58. setlength{oddsidemargin}{-69pt}
  59. begin{document}$$F^{m}$$end{document}-transform with application (pp. 32-52) https://doi.org/10.1016/j.fss.2017.12.008